KABUL—Afghanistan’s defense chief said Monday the country’s U.S.-backed security forces were struggling to counter a Taliban offensive on a strategic eastern city, a departure after days of rosy assessments about progress in dislodging the insurgents.

The insurgents’ siege on the city of Ghazni, where local officials and residents have described clashes at key government outposts, dire humanitarian conditions and militants freely roaming the streets, threatened a fledgling peace process and raised questions about the progress...